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CHAP. X] SUM AND NuMBBK OP DlVISOBS. 305
sum $k(x) of the Mh powers of the divisors > \/# of x over the sum of the Mh powers of the remaining divisors, it follows at onee that
with a similar formula for S^(aj) , where ^(2) is the excess of \f/kf(x) over the sum of the Mh powers of the divisors < •%/# of x. For k = 1, ttte last formula reduces to the third one of Hermite's.
Let Xk(%) be the sum of the Mh powers of the odd divisors of x; Xk'(x) that for the odd divisors > \/x; X* (x) the excess of the latter sum over the sum of the Mh powers of the odd divisors < \fx of x] Xk'"(x) the excess of the sum of the Mh powers of the divisors 8s±l>v5 of x over the sum of the Mh powers of the divisors 8s=*=3<\/5 of x. For y=2x and y = 2x— 1, the sum from x = l to x = n of Xk(y), Xk'(y)> Xk" (y) and Xk'"(y) are expressed as complicated sums involving the functions Sk and [x].
E. Pfeiffer90 attempted to prove a formula like (7) of Dirichlet,17 where now eis 0(nl/z+k) for every &>0. Here Og(T) means a function whose quotient by g(T) remains numerically less than a fixed finite value for all sufficiently large values of T. E. Landau91 noted that the final step in the proof fails from lack of uniform convergence and reconstructed the proof to secure this convergence.
L. Gegenbauer,92 in continuation of his80 paper, gave similar but longer expressions for
S T(T/), £ er*(y) (y=4c+l, 6z+l, 8x+3, 8x+5, 8x+7)
and deduced similar tests for the primaHty of y.
Gegenbauer920 found the mean of the number of divisors Xrc+a of a number of s digits with a complementary divisor M^+j3; also for divisors
Gegenbauer926 evaluated A (1)4- . . . +A(ri) where A(x) is the sum of the pth powers of the <rth roots of those divisors d of x which are exact crth powers and whose complementary divisors exceed kdr/*. A special case gives (11), p. 284 above.
Gegenbauer920 gave a formula involving the sum of the Mh powers of those divisors of 1, . . . , m whose complementary divisors are divisible by no rth power >1.
90Ueber die Periodicitat in der Teilbarkeit . . ., Jahresbericht der Pfeiffer'scben Lehr- und Erzieh-
ungs-Anstalt zu Jena, 1885-6, 1-21. "Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ha, 1912, 2195-2332; 124, Ha, 1915, 469-550.
"Ibid., 93, II, 1886, 447-454.
"«Sitzungsber. Ak. Wiss. Wien (Math.), 93, 1886, II, 90-105. &«., 94, 1886, II, 35-40.